By the sounds of it, I suspect that your iPhone is connecting to your car's Bluetooth audio. By default, the navigation audio prompts will play over Bluetooth audio, i.e. over your car's auxiliary Bluetooth audio (music) source (e.g. if you have AM/FM or satellite radio playing, the navigation prompts won't be heard over these audio modes).
When navigating I typically have my iPhone playing music over Bluetooth, and the nav prompts will chime in and override what's playing. But if I'm using any other audio source on my stereo, the nav prompts are not available since they're on the Bluetooth audio stereo connection. So try again but while playing some form of audio from your iPhone over Bluetooth, you'll likely get the prompts.
Depending on the car stereo system, there are ways to change your Bluetooth connection to "Hands Free Profile" (HFP), which means your nav prompts will be handled like phone calls -- chiming in no matter what's playing on your stereo (even nothing at all). But in my experience this can get really annoying, as each prompt is followed by the stereo's built-in voice prompt such as "Call completed."
Play around with it, and read this article for reference:
No sound with Maps while driving
